This article introduces the formal definition of a group in Abstract Algebra and explains the four fundamental group axioms: closure, associativity, identity, and inverse. Through clear examples and complete proofs, the reader will understand why sets such as ℝ, ℤ, ℚ, and ℂ form groups under addition or multiplication. After reading, students will be able to verify group structures, recognise identity and inverse elements, and apply the axioms to new algebraic systems. This serves as a foundational step for further study of subgroups, cyclic groups, and homomorphisms.

Let $G$ be a non-empty set and $\circ$ be a binary composition. $G$ is said to be a group if 1. $x \circ y \in G~ \forall \, x,y \in G$. 2. $x \circ \left(y \circ z\right)=\left(x \circ y \right) \circ z~ \forall \, x,y,z \in G$. 3. There exists an element $e\in G$, such that $x \circ e=e\circ x=x~\forall \, x \in G$. Here $e$ is said to the identity element. 4. For each element $x\in G$, there exists an element $y\in G$ such that $x \circ y=y\circ x=e.$ Here $y$ is said to the inverse of $e$ and is denoted by $x^{-1}$.